Certainly! Let's simplify the given expression step-by-step.
The given expression is:
[tex]\[ 14 \left[2(x+y)^4\right]^7 \][/tex]
### Step 1: Simplify the Inner Expression
First, look at the expression inside the brackets:
[tex]\[ 2(x+y)^4 \][/tex]
### Step 2: Apply the Exponent
We need to raise everything inside the brackets to the power of 7:
[tex]\[ \left[2(x+y)^4\right]^7 \][/tex]
Using the exponentiation rule [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex], we can separate the constant and the polynomial terms:
[tex]\[ \left[2(x+y)^4\right]^7 = 2^7 \cdot \left[(x+y)^4\right]^7 \][/tex]
### Step 3: Simplify the Exponent
Simplify the separated terms:
[tex]\[ \left[(x+y)^4\right]^7 \][/tex]
Using the power of a power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we get:
[tex]\[ \left[(x+y)^4\right]^7 = (x+y)^{4 \cdot 7} = (x+y)^{28} \][/tex]
So the expression is now:
[tex]\[ 2^7 \cdot (x + y)^{28} \][/tex]
### Step 4: Calculate the Constant Term
Now calculate the numerical value of [tex]\(2^7\)[/tex]:
[tex]\[ 2^7 = 128 \][/tex]
Thus, the expression becomes:
[tex]\[ 128 (x + y)^{28} \][/tex]
### Step 5: Multiply by the Outer Constant
Finally, multiply this result by the outer constant, 14:
[tex]\[ 14 \cdot 128 \cdot (x + y)^{28} \][/tex]
Calculate the constant multiplication:
[tex]\[ 14 \cdot 128 = 1792 \][/tex]
So the expression simplifies to:
[tex]\[ 1792 (x + y)^{28} \][/tex]
### Final Result
The simplified form of the given expression is:
[tex]\[ 1792 (x + y)^{28} \][/tex]
